Fixed points for branched covering maps of the plane

2019/06/10 by Alejo García, García, Alejo
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1906.03770

openalex publication_date 2019/06/10 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

A well-known result from Brouwer states that any orientation preserving homeomorphism of the plane with no fixed points has an empty non-wandering set. In particular, an invariant compact set implies the existence of a fixed point. In this paper we give sufficient conditions for degree 2 branched covering maps of the plane to have a fixed point, namely: A totally invariant compact subset such that it does not separate the critical point from its image An invariant compact subset with a connected neighbourhood U, such that Fill(U ∪ f(U)) does not contain the critical point nor its image. An invariant continuum such that the critical point and its image belong to the same connected component of its complement.

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